Neural Network Virtual Sensors

• Neural network virtual sensors are data-driven models that infer critical system parameters by mapping available sensor data, serving as cost-effective alternatives to physical sensors.
• They employ diverse architectures—including feedforward, CNN, GNN, and operator networks—with training paradigms like physics-informed losses and adaptive tuning to ensure robust, real-time performance.
• Applications span automotive, industrial IoT, digital twins, and health monitoring, enabling enhanced monitoring, reduced hardware costs, and agile system integration.

Neural network virtual sensors are data-driven models designed to infer critical system parameters or states by learning mappings from available sensor data, thus minimizing reliance on additional physical instrumentation. These virtual sensors exploit the nonlinear modeling capabilities of neural networks, enabling precise estimation or prediction of quantities that are either impractical or prohibitively expensive to measure directly. Applications extend across domains such as automotive engines, industrial automation, energy systems, virtual reality, and health monitoring, with increasing adoption where cost, reliability, and extensibility are central concerns.

1. Neural Architectures for Virtual Sensor Modeling

Neural network virtual sensors are implemented predominantly with architectures attuned to the nature of the underlying signals and system complexity:

• Feedforward Neural Networks: Used in virtual sensor modeling for diesel engines, as in a three-layer configuration with nonlinear activation (tansig) in the hidden layer and a linear output layer for regression outputs. Training variants include Levenberg–Marquardt and Bayesian regularization backpropagation, tailored to capture nonlinear dependencies while providing low-complexity models faithful to real-time engine controller constraints (Rastogi et al., 2017).
• Deep Convolutional Architectures: Applied to modalities such as virtual IMU data for human activity recognition, deep CNNs—augmented with unsupervised deconvolutional layers—facilitate joint feature learning and denoising. Training involves composite loss functions balancing supervised task objectives with unsupervised reconstruction penalties, thereby enhancing robustness and generalization to diverse sensor signatures (Xiao et al., 2020).
• Graph Neural Networks (GNNs): For systems with spatially distributed or heterogeneous sensors (e.g., bearing load prediction), GNNs and, more specifically, Heterogeneous Temporal Graph Neural Networks (HTGNNs) model inter-sensor dependencies and multi-scale temporal characteristics. Distinct encoders (e.g., GRUs for low-frequency, CNNs for high-frequency nodes) are deployed per sensor type, with message passing mechanisms (Graph Convolution/Attention Networks) reflecting both modality-specific and cross-modality interactions (Zhao et al., 2 Apr 2024, Zhao et al., 26 Jul 2024).
• Operator Neural Networks: Deep operator networks (DeepONet) and Multi-Input Operator Networks (MIONet) generalize conventional NNs to learn mappings between function spaces, allowing virtual sensors to infer full-field spatial distributions from scalar or profile input functions. The dual-branch (input) and trunk (coordinate) architecture efficiently fuses operational data and spatial information, enabling rapid, accurate virtual measurements across inaccessible domains (Hossain et al., 17 Oct 2024, Kobayashi et al., 28 Nov 2024).
• Hybrid and Adaptive Observer Architectures: Integration of neural networks with adaptive control (e.g., Sliding Mode Control, SMC) produces robust software sensors for nonlinear systems. Here, a neural network parameterizes time-varying observer gains, working synergistically with SMC to enforce estimation error convergence under model uncertainties, disturbances, and measurement limitations. Training is physics-informed, constrained by the system’s governing equations (Farkane et al., 9 Jul 2025).

2. Training Paradigms and Performance Optimization

The effectiveness of neural network virtual sensors relies on systematic training methodologies and hyperparameter optimization techniques:

• Adaptive Weights and Biases Tuning: Iterative coefficient-based search algorithms adjust neural network weight/bias initializations, employing divide-and-conquer strategies and multi-stage grid refinement. Performance is judged via metrics (e.g., mean squared error, regression coefficient, range, countPercent), with thresholds derived from confidence intervals over performance distributions (Rastogi et al., 2017).
• Transfer and Domain Adaptation: When trained on synthetic or virtual datasets, fine-tuning is performed on smaller datasets of real data to bridge domain gaps. For example, convolutional models pretrained on virtual IMU data are refined using real IMU signals by updating only high-level layers, exploiting the learned generic features while adapting to the statistics and noise of real sensors (Xiao et al., 2020).
• Physics-Informed Augmentation: GNN architectures are enhanced with “augmented nodes” computed via domain-specific relations (e.g., Bernoulli and Darcy-Weisbach equations in district heating systems), fusing empirical sensor data and physically modeled variables to improve virtual sensor reliability under sparse or noisy measurements (Niresi et al., 11 Apr 2024).
• Robust and Certifiable Training: Duality-based robust optimization techniques yield networks with provable performance specifications under bounded adversarial or stochastic input perturbations, a critical property in regulatory- or safety-critical contexts (e.g., fuel injection systems). MILP-based exact verification is used to empirically certify worst-case performance guarantees (Wong et al., 2020).

3. Application Domains and System Integration

Neural network virtual sensors are deployed in a range of real-world systems:

Domain	Target Variables	NN Paradigm/Approach
Diesel/Combustion Engines	Oil pressure, fuel injection	Feedforward NN; duality-based robust NN
Industrial IoT	Pressure, temperature, flow rate	Physics-enhanced GNNs
Prognostics/PHM	Bearing loads	Heterogeneous temporal GNNs
Digital twins (nuclear)	Pressure, velocity, turbulence	DeepONet, MIONet
Robotics/NMPC	End-effector trajectories	Multi-stage NN with temporal loss
Autonomous vehicles	Sideslip angle, vehicle states	Hybrid (Informer+motion model)
Wearables/XR	IMU signals, HAR input	Deep CNN (for HAR); signal processing in XR

Virtual sensors support critical tasks including condition monitoring, health management, trajectory tracking in robotics, and secure biometrics (e.g., using synthesized side channels for multimodal authentication) (Long et al., 2023). They also enable advanced testing for autonomy via neural simulation of sensor outputs under manipulated or rare scenarios (UniSim) (Yang et al., 2023).

4. Performance Metrics, Validation, and Robustness

Assessment of neural network virtual sensors is multi-dimensional:

• Predictive Accuracy: Quantified via mean squared error (MSE), mean absolute error (MAE), mean absolute percentage error (MAPE), F1-score, R², etc., depending on regression or classification task. Typical improvements after adaptive tuning are evidenced by MSE reduction and increased accuracy clustering (Rastogi et al., 2017, Zhao et al., 2 Apr 2024).
• Robustness and Reliability: Certified worst-case relative errors under bounded input perturbations or adversarial noise (e.g., maximum MRE ≤ 16.5% for robust fuel injection virtual sensors, with empirical confirmation by MILP verification) (Wong et al., 2020).
• Real-Time and Computational Efficiency: Architectures (e.g., DeepONet, MIONet) enable inference latencies orders of magnitude below conventional CFD or physics-based simulations (e.g., 0.135 s vs. 200 s), critical for real-time deployment in digital twins and closed-loop control (Hossain et al., 17 Oct 2024, Kobayashi et al., 28 Nov 2024). Complexity-reduction techniques, such as lightweight predictors and compressed feature maps, ensure feasibility for embedded and resource-constrained environments (Masti et al., 2021).
• Adaptability: Transfer learning schemes permit rapid adaptation to new sensor deployments or domains with minimal retraining effort, supporting long-term scalability in dynamic industrial or sensor-replacement environments (Muñoz-Molina et al., 2021).

5. Theoretical Guarantees and Physics Constrained Learning

Rigorous convergence, robustness, and generalization are central to the acceptance of virtual sensors in high-stakes applications:

• Physics-Informed Losses: Integration of governing differential equations as constraints during training (e.g., in observer design for nonlinear systems) ensures that neural models remain consistent with known physical laws, even in the absence of full state ground truth (Farkane et al., 9 Jul 2025).
• Lyapunov-Based Convergence Analyses: Sufficient conditions for exponential estimation error convergence are derived using Lyapunov candidate functions and Riccati inequalities, incorporating approximation error decay and time-varying observer gains (Farkane et al., 9 Jul 2025).
• Operator Learning and Generalization: Operator network frameworks generalize mappings from input function spaces to output function spaces, with demonstrated resilience against dataset shift and the need for continuous retraining, enabling deployment under varying operational regimes (Hossain et al., 17 Oct 2024, Kobayashi et al., 28 Nov 2024).

6. Cost, Deployment, and Industrial Impact

Neural network virtual sensors often replace hardware-intense measurement chains:

• Cost Reduction: By learning to infer key measurements from available, lower-cost sensor modalities, virtual sensors eliminate the need for widespread deployment of expensive sensors, yielding simplified designs and streamlined ECMs, particularly crucial in fields such as automotive powertrains (Rastogi et al., 2017).
• Software Updates and Adaptability: Algorithmic updates, model calibration, and adaptation can often be performed entirely in software, permitting agile response to regulatory changes, operational reconfiguration, or sensor faults without physical modification.
• Integration with Digital Twins: As a core digital twin component, virtual sensors boost the spatiotemporal resolution and fidelity of monitoring and control, crucial for PHM (Prognostics and Health Management), predictive maintenance, and safety-critical system management in energy and beyond (Hossain et al., 17 Oct 2024, Kobayashi et al., 28 Nov 2024).
• Limitations and Cautions: Uniform sensor drifts may go undetected if the model learns only relative behaviors (Muñoz-Molina et al., 2021); spectral bias may reduce accuracy for high-frequency dynamics (Hossain et al., 17 Oct 2024); domain adaptation effectiveness depends on similarity of new environments to training conditions.

7. Outlook and Future Research Directions

Several future research avenues are underscored:

• Enhanced Robustness: Stronger certification, e.g., via adversarial robustness or rigorous uncertainty quantification, remains a priority for regulated domains.
• Hybrid/Physics-Aware Models: Deeper integration of physics priors and domain knowledge (e.g., region-specific neural operators, hybrid physics–data schemes) addresses limitations of purely data-driven approaches in extrapolative settings.
• Generalization Across Domains: Transfer learning, dynamic graph construction, and adaptive message-passing mechanisms promise broader applicability of GNN-based virtual sensors to new modalities and industrial contexts (Zhao et al., 26 Jul 2024).
• Human–Machine Interaction: Innovations in XR and digital twinning—where virtual sensors are visualized, designed, and debugged in immersive environments—enable rapid prototyping and improved end-user interpretability of virtual sensor data streams (Liang et al., 10 Sep 2024).

These trends indicate that neural network virtual sensors will play an expanding role in future autonomous, data-driven, and cyber-physical systems, providing an adaptable interface between sensed data and actionable insights.